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26 II RIAL NUMBERS
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Let a = «p/(*), * = «?*<*)>• then /W < fl and 9(x} < * for every
)+^)<i+*.and the ^ inequa"t5: fo"T; ^Let
; then for every xeA, /(*) + c </(*) +^) < rf =
); but this yields f(x)^d-c for every *eA, hence - c, or * + c < 4 which is the second inequality. |
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(2.3.10) £er/Z»e fl majorized mapping of A. into R; then, for every real number
C, Slip (/(JC) + €) = €
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Take for # the constant function equal to c in (2.3.9).
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(23.11) L?//i (resp./2) fe a majorized mapping of A1 (resp. A2) w/o R;
) w majorized, and
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sup (/jfe) +/2fe)) = sup /jfe) + sup /2(x2).
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A2
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Apply (2.3.7) and (2.3.10).
We leave to the reader the formulation of the similar properties for inf
(change the signs everywhere). |
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PROBLEM
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Let *->!(#) be a mapping of R into the set of open intervals of R, such that I(jc) be an
open interval of center x and of length ^ c (c being a given number > 0). Show that, for eveiy closed interval [a, b] of R, there exist a finite number of points xt of [a, b] such that: (1) the intervals l(xt) form a covering of [a, b]; (2) the sum of the lengths of the !(*/) is < c + 2(b-a). (Prove that if the theorem is true for any interval [a, x] such that a ^ x < u < b, then there exists v such that u < v < b and that the theorem is still true for any interval [a, y] such that a ^ y < v. Consider then the l.u.b. of all numbers u < b such that the theorem is true for any interval [a, x] such that a < x < u.) Show by an example that the majoration is best possible. |
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